On the minimum vertex cover of generalized Petersen graphs

Discrete Applied Mathematics xxx (xxxx) xxx

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On the minimum vertex cover of generalized Petersen graphs✩ Dannielle D.D. Jin a , David G.L. Wang a,b , a b

∗

School of Mathematics and Statistics, Beijing Institute of Technology, 102488 Beijing, PR China Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, 102488 Beijing, PR China

article

info

Article history: Received 11 May 2017 Received in revised form 10 December 2018 Accepted 20 December 2018 Available online xxxx Keywords: Generalized Petersen graph Independent set Minimum vertex cover

a b s t r a c t It is known that any vertex cover of the generalized Petersen graph P(n, k) has size at least n. Behsaz, Hatami and Mahmoodian characterized such graphs with minimum vertex cover numbers n and n + 1, and those with k ≤ 3. For k ≥ 4 and n ≥ 2k + 2, we prove that if the 2-adic valuation of n is less than or equal to that of k, then the minimum vertex cover number of P(n, k) equals n + 2 if and only if n ∈ {2k + 2, 3k − 1, 3k + 1}. © 2018 Elsevier B.V. All rights reserved.

1. Introduction A vertex cover of a graph is a set of vertices that touch all edges of the graph, that is, a vertex subset whose complement is an independent set. In this paper, we consider the size of a minimum vertex cover of generalized Petersen graphs, which were introduced by Watkins [22] in the consideration of a conjecture of Tutte [21]. Definition 1.1. Let n ≥ 3 and 1 ≤ k < n/2. The generalized Petersen graph, denoted P(n, k), is defined to be the graph with vertex set {uj , vj : j ∈ Zn } and edge set {uj uj+1 , vj vj+k , uj vj | j ∈ Zn }, where Zn is the cyclic group of integers modulo n. For example, the graph P(4, 1) is the cube skeleton, and P(5, 2) is the Petersen graph. It was Coxeter [7] who firstly studied the graphs P(n, k) with coprime parameters n and k. Many graph theoretic properties and algorithmic properties of the generalized Petersen graphs have been investigated; see for instance [1,6,8,9,11–20]. Besides the minimum vertex cover problem, some other difficult problems concerning generalized Petersen graphs have also received considerable attention. For example, Behzad, Behzad, and Praeger [3] showed that ⌈3n/5⌉ is an upper bound for the domination number of the graph P(n, k) when n is odd; see also [23]. In this paper, we characterize the set of generalized Petersen graphs P(n, k) whose minimum vertex cover number is n + 2, satisfying the property that the 2-adic valuation of n is less than or equal to that of k. In Section 2 we recall some known results and preliminaries about the structure of any minimum vertex cover of a generalized Petersen graph. In Section 3, we prove the main result Theorem 3.15 using elementary number theory. ✩ This paper was supported by General Program of National Natural Science Foundation of China (Grant No. 11671037), the Fundamental Research Funds for the Central Universities (Grant No. 20161742027), and the Beijing Institute of Technology Research Fund Program for Young Scholars (Grant No. 2015CX04016). ∗ Corresponding author at: School of Mathematics and Statistics, Beijing Institute of Technology, 102488 Beijing, PR China. E-mail addresses: [email protected] (D.D.D. Jin), [email protected] (D.G.L. Wang). https://doi.org/10.1016/j.dam.2018.12.011 0166-218X/© 2018 Elsevier B.V. All rights reserved.

Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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2. Preliminaries For any graph G, we denote by β (G) the size of a minimum vertex cover of G. For convenience, we denote β (n, k) = β (P(n, k)). The exact value of β (n, k) is known for k ∈ {1, 2, 3, 5}; see [2,4,24]. Theorem 2.1. The following formulas hold:

β (n, 1) =

n,

if n is even,

n + 1,

if n is odd;

{

β (n, 2) = ⌈6n/5⌉; { n, β (n, 3) = n + 2, { n, β (n, 5) = n + 3,

if n is even, if n is odd; if n is even, if n is odd.

Note that the number β (n, k) is at least n, since for any minimum vertex cover W of P(n, k), every vertex ui such that vi ̸∈ W is a member of W . Behsaz, Hatami, and Mahmoodian [2] characterized the generalized Petersen graphs with β (n, k) = n and those with β (n, k) = n + 1. Theorem 2.2. The equality β (n, k) = n holds if and only if n is even and k is odd. The equality β (n, k) = n + 1 holds if and only if either (i) n is odd and k = 1, or (ii) n = 5 and k = 2. In order to determine the pairs (n, k) with β (n, k) = n + 2, we first investigate the structure of the graph P(n, k). We denote the induced subgraphs P(n, k)[u0 , u1 , . . . , un−1 ] and

P(n, k)[v0 , v1 , . . . , vn−1 ],

by O and I , respectively. These are called the outer cycle and the graph of inner cycle(s), respectively. The graph I of inner cycles consists of d = gcd(n, k) cycles of the same order n/d. It is clear that integer n/d is odd if and only if ν2 (n) ≤ ν2 (k), where the notation ν2 (m) stands for the 2-adic valuation of an integer m, viz. the maximum integer h such that 2h divides m. A number of upper bounds for β (n, k) were obtained by explicit constructions of minimum vertex covers, see [2,10]. For instance,

β (n, k) ≤

{

(5n + d)/4,

if ν2 (n) ≤ ν2 (k);

5n/4,

otherwise.

However, few lower bounds are known. Behsaz et al. [2, Proposition 2] obtained that

β (n, k) ≥ n + ⌈d/2⌉,

if n is odd.

We now generalize this slightly by weakening the oddness of n to that ν2 (n) ≤ ν2 (k), by a small modification of their proof; see Lemma 2.5. For a graph G with a vertex subset W and a subgraph H, we say that W minimally covers H if (i) every component of H is either a path or a cycle, and (ii) for any component H ′ of H, the set W ∩ H ′ is a minimum vertex cover of H ′ . The structure of any minimum vertex cover of a path and that of a cycle are presented in Lemmas 2.3 and 2.4, respectively. We omit their proofs for they can be shown by easy combinatorial arguments. Lemma 2.3. Let W be a vertex cover of a path P1 P2 · · · Pℓ . Then the following statements are equivalent: (i) W is a minimum vertex cover. (ii) |W | = ⌊ℓ/2⌋. (iii) Either W = {P2i : 1 ≤ i ≤ ⌊ℓ/2⌋} or W = {Pℓ+1−2i : 1 ≤ i ≤ ⌊ℓ/2⌋}. □ Lemma 2.4. Let W be a vertex cover of a cycle C1 C2 · · · Cℓ . Then the following statements are equivalent: (i) W is a minimum vertex cover. (ii) |W | = ⌈ℓ/2⌉. (iii) If ℓ is even, then either W = {C2i : 1 ≤ i ≤ ℓ/2} or W = {C2i−1 : 1 ≤ i ≤ ℓ/2}; if ℓ is odd, then there is a unique 1 ≤ k ≤ ℓ such that W = {Ck } ∪ {Ck+2i−1 : 1 ≤ i ≤ (ℓ − 1)/2}, where the subscripts are considered modulo ℓ. □ Lemma 2.5. Suppose that ν2 (n) ≤ ν2 (k). Let W be a vertex cover of the graph P(n, k). Then the following hold: (i) |W ∩ V (I )| ≥ ⌊n/2⌋ + ⌈d/2⌉, where the equality holds if and only if W minimally covers I ; (ii) |W | ≥ n + ⌈d/2⌉, where the equality holds if and only if W minimally covers each of O and I . Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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Proof. By Lemma 2.4, we deduce that

|W ∩ V (I )| ≥ ⌈n/(2d)⌉d

(2.1)

= ⌊n/2⌋ + ⌈d/2⌉,

(2.2)

where Eq. (2.2) is true since ν2 (n) ≤ ν2 (k). It is clear that the equality in Eq. (2.1) holds true if and only if W minimally covers I . By Lemma 2.4, |W ∩ V (O)| ≥ ⌈n/2⌉. Since |W | = |W ∩ V (O)| + |W ∩ V (I )|, we obtain (ii) immediately. □ Here is a key concept for understanding the structure of a minimum vertex cover of P(n, k). Definition 2.6. For any set S of vertices of the graph P(n, k), define Σ (S) to be the size of a minimum vertex cover of the subgraph O[{ui : vi ∈ S }]. Lemma 2.7. Let W be a vertex cover of the graph P(n, k). Then |W | ≥ n + Σ (W ), where the equality holds if and only if W is a minimum vertex cover. Proof. Write W ′ = {ui ∈ O: vi ∈ W }. It is clear that |W | = n + |W ∩ W ′ |. Since each component of the subgraph O[W ′ ] is a path, the intersection W ∩ W ′ covers O[W ′ ]. It is immediate from the definition that |W ∩ W ′ | ≥ Σ (W ), where the equality holds if and only if W is a minimum vertex cover. □ Recall from Boben, Pisanski, and Žitnik [5] that isomorphic generalized Petersen graphs are classified by the equivalence relation P(n, h) ∼ = P(n, k) ⇐⇒ hk ≡ ±1

(mod n).

(2.3)

It follows that P(2k + 1, k) ∼ = P(2k + 1, 2). In view of Theorem 2.1, we shall assume throughout the rest of this paper that n ≥ 2k + 2

and

k ≥ 4.

For any integers a and b, we denote

[ a, b ] =

{

{a, a + 1, . . . , b}, if a ≤ b; ∅, if a > b.

Denote by mod(n, p) the least nonnegative residue of n modulo p. For any integer set S, we denote mod(S , p) = {mod (s, p): s ∈ S }. We further adopt the Minkowski addition notation S + a = {s + a: s ∈ S }. 3. Characterization for β(n, k) = n + 2 with odd n/gcd(n, k) The goal of this section is to establish Theorem 3.15, a characterization of the generalized Petersen graphs P(n, k) with minimum vertex cover number n + 2, provided that ν2 (n) ≤ ν2 (k). Our argument is based on the following consequence of Lemmas 2.5 and 2.7. For any vertex cover W of a graph G, we say that W nearly minimally covers G if any minimum vertex cover of G has size |W | − 1. Lemma 3.1. Let k ≥ 4 and n ≥ 2k + 2. If ν2 (n) ≤ ν2 (k) and β (n, k) = n + 2, then d ∈ {1, 2}. Moreover, every minimum vertex cover of the graph P(n, k), except that of P(11, 4), minimally covers each inner cycle, and nearly minimally covers the outer cycle. Proof. Since β (n, k) = n + 2, Lemma 2.5 implies d ≤ 4. Let C be a minimum vertex cover. Then |C | = n + 2 and Σ (C ) = 2 by Lemma 2.7. Assume that d ∈ {3, 4}. Then by Lemma 2.5(ii), the cover C minimally covers O. As a result, we can suppose without loss of generality that C ∩ V (O) = {u0 , u2 , . . . , u2⌈n/2⌉−2 }.

(3.1)

Since every edge ui vi is covered by C , we find

{v1 , v3 , . . . , v2⌊n/2⌋−1 } ⊂ C .

(3.2)

Denote by I1 the inner cycle containing the vertex v1 . Note that if k is odd, then d = 3 and n is odd. Now let j = 1 + k⌊n/k⌋. Since C covers the edge vj vj+k , either vj ∈ C or vj+k ∈ C . Consider the set S of two edges in I1 defined by

⎧ {v1 v1+k , ⎪ ⎪ ⎨ {v1 v1−k , S= ⎪{v1 v1−k , ⎪ ⎩ {v1 v1−k ,

v1+k v1+2k }, vj vj+k }, vj vj−k }, vj+k vj+2k },

if k is even; if k is odd and j is odd; if k is odd, j is even, and vj ∈ C ; if k is odd, j is even, and vj+k ∈ C .

Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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In view of (3.2), both ends of each edge in S are in C . But this is impossible since C minimally covers I1 by Lemma 2.5(ii). This proves d ̸ ∈ {3, 4}. Below we can suppose that d ∈ {1, 2}. Assume that |C ∩ V (I )| ≥ ⌊n/2⌋ + 2. By Lemma 2.4, we obtain n + 2 = |C | = |C ∩ V (O)| + |C ∩ V (I )| ≥ ⌈n/2⌉ + ⌊n/2⌋ + 2 = n + 2. It follows that |C ∩ V (O)| = ⌈n/2⌉ and |C ∩ V (I )| = ⌊n/2⌋ + 2. The former equality allows us to suppose without loss of generality that Eq. (3.1) holds, and consequently (3.2) holds. To sum up, we obtain T ⊂ C , where T = {u0 , u2 , . . . , u2⌈n/2⌉−2 , v1 , v3 , . . . , v2⌊n/2⌋−1 }. Note that |C | − |T | = 2. However, if k is odd, then n is odd by Theorem 2.2. In this case, the three edges vi vi−k for i = 0, 2, 4 are not covered by T , a contradiction. If k ≥ 6 is even, then the three edges vi vi+k for i = 0, 2, 4 are not covered by T and we obtain a contradiction again. Therefore, we infer that k = 4. If n is even, then no edge on the inner cycle containing v0 is covered by T . It follows that ⌈n/4⌉ ≤ 2 and n ≤ 8, contradicting the assumption n ≥ 2k + 2. Thus n is odd. If n ≥ 13, then the three edges v0 v4 , v2 v6 and v8 v12 are not covered by T , a contradiction. To summarize, we obtain k = 4 and n = 11. Suppose that the graph P(n, k) is not P(11, 4). Then the above arguments confirm |C ∩ V (I )| ≤ ⌊n/2⌋+ 1. By Lemma 2.5(i), we conclude that |C ∩ V (I )| = ⌊n/2⌋ + 1 and that C minimally covers each inner cycle. As a consequence, we obtain |C ∩ V (O)| = |C | − |C ∩ V (I )| = ⌈n/2⌉ + 1, i.e., C nearly minimally covers the outer cycle. □ Next, we will deal with the cases d = 1 and d = 2 in Sections 3.1 and 3.2 respectively. The conclusion is in Section 3.3. 3.1. Case d = 1 We only need to consider the case n is odd by virtue of Theorem 2.2. Theorem 3.2. Let k ≥ 4. Let n ≥ 2k + 2 be an odd integer. Suppose that gcd(n, k) = 1. Let k′ be the smallest positive multiplicative inverse of 2k modulo n, namely 2kk′ ≡ 1 (mod n) and k′ < n. For any vertex cover W that minimally covers I ,

Σ (W ) =

{

k0 p/2,

if p is even,

(n + 1 − k0 p − k0 )/2, if p is odd,

where k0 = min(k′ , n − k′ ) and p = ⌊(n + 1)/(2k0 )⌋. Proof. From the definition, one infers that p ≥ 1. Consider the sequence S1 , S2 , . . . of subsets of Zn defined by S1 = [0, (n − 1)/2] and the recurrence Sr +1 = Sr ∩ (Sr + k′ ),

for r ≥ 1.

(3.3)

By using elementary number theory, one may deduce that

{ [(r − 1)k′ , (n − 1)/2], if k′ < n/2; Sr = ′ [0, (n − 1)/2 − (r − 1)(n − k )], if k′ > n/2. Let W be a vertex cover which minimally covers I . Without loss of generality, we can suppose that W ∩ I = {v2ki : i = 0, 1, . . . , ⌈n/2⌉ − 1}. Let i, j ∈ Sr for some r ≥ 1. By Eq. (3.3), we have the equivalences 2ki ≡ 2kj − 1

(mod n)

⇐⇒

j ≡ i + k′

(mod n)

⇐⇒

j ∈ Sr +1 .

Thus for any j ∈ Sr , the r consecutive vertices of the inner cycle with subscripts in the set [2kj − r + 1, 2kj] belong to the cover W . As a consequence, every component of the subgraph O[{ui : vi ∈ W }] has order p + 1 or p. Moreover, the number of components of order p + 1 is |Sp+1 | = (n + 1)/2 − pk0 . It follows that

Σ (W ) = |Sp+1 | · ⌊(p + 1)/2⌋ + (k0 − |Sp+1 |) · ⌊p/2⌋, which simplifies to the desired formula. □ Taking the graph P(11, 4) for example, we can calculate that k′ = 7, k0 = 4, p = 1 and Σ (W ) = 2. In general, in the case that k = 4, Theorem 3.2 reduces to Corollary 3.3. Corollary 3.3. Let n ≥ 11 be an odd integer. For any minimum vertex cover W that minimally covers I (n, 4),

Σ (W ) =

{

⌊(n + 1)/4⌋, if n ≡ ±1 ⌈n/8⌉, if n ≡ ±3

(mod 8); (mod 8).

For the previous example P(11, 4), we find Σ (W ) = ⌈11/8⌉ = 2 again. Theorem 3.4. Let k ≥ 4. Suppose that n ≥ 2k + 3 is odd. If gcd(n, k) = 1, then β (n, k) = n + 2 if and only if n ∈ {3k − 1, 3k + 1}. Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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Proof. Since k ≥ 4, we obtain n ≥ 11. Let C be a minimum vertex cover of the graph P(n, k). By Lemma 2.7, we find β (n, k) = n + 2 if and only if Σ (C ) = 2. Let us retain the notation in Theorem 3.2. If n ∈ {3k ± 1}, then k′ ∈ {(n ∓ 3)/2}. It follows that k0 = (n − 3)/2 and p = 1. By Theorem 3.2, we obtain Σ (C ) = 2. Conversely, suppose that Σ (C ) = 2 and n ̸ ∈ {3k ± 1}. Then n ≥ 17. By Lemma 3.1, the cover C minimally covers the inner cycle, which allows us to use Theorem 3.2. If p is even, then k0 p/2 = Σ (C ) = 2 and p = ⌊(n + 1)/(2k0 )⌋ = ⌊(n + 1)p/8⌋ > p, a contradiction. Thus p is odd, and (n + 1 − k0 p − k0 )/2 = Σ (C ) = 2, i.e., n − (p + 1)k0 = 3. Together with the definition of p, we find p = 2 · ⌊2/k0 ⌋ + 1 and thus n = 2k0 + 3 + 2k0 ⌊2/k0 ⌋. Since n ≥ 17, we find k0 ≥ 7 and n = 2k0 + 3. It follows that

{ n=

3k + 1, if k0 = k′ ; 3k − 1,

if k0 = n − k′ .

This completes the proof. □ 3.2. Case d = 2 For the rest of this subsection, we suppose that d=2

and n ≡ 2

(mod 4).

The condition n ≡ 2 (mod 4) comes from the assumption that ν2 (n) ≤ ν2 (k). In this case, there are exactly two inner cycles. In other words, I = I0 ∪ I1 . Let C be a minimum vertex cover of the graph P(n, k). By Lemma 3.1, we can suppose, without loss of generality, that for j = 0,1, C ∩ Ij = {v2ki+jt : 0 ≤ i ≤ (n − 2)/4} for some odd integer t. Definition 3.5. Let k ≥ 4 and n ≥ 2k + 2. Suppose that n ≡ 2 (mod 4) and gcd(n, k) = 2. For any odd integer t, define Σn (t) = Σ (C ∩ I ), that is,

Σn (t) = Σ ({v0 , v2k , . . . , v(n/2−1)k , vt , vt +2k , . . . , vt +(n/2−1)k }). In light of Lemma 2.7, we need to compute the minimum value of the function Σn (t). Lemma 3.6. Σn (t) = Σn (−t) for any positive odd integer t ≤ n/2. Proof. Let s = (n − 2)k/4 and Λ = mod {0, 2k, . . . , (n/2 − 1)k}, n . Then

(

s + m ∈ Λ ⇐⇒ s − m ∈ Λ,

)

for any m ∈ Z.

As a consequence, we derive that s + m + t ∈ Λ + t ⇐⇒ s − m − t ∈ Λ − t . Note that for all m, m′ ∈ Z,

|(s + m) − (s + m′ + t)| = 1 ⇐⇒ |(s − m) − (s − m′ − t)| = 1. In other words, any two elements s + m and s + m′ + t in the set Ωn (t) = Λ ∪ (Λ + t) are consecutive modulo n if and only if the elements s − m and s − m′ − t in the set Ωn (−t) are consecutive modulo n. On the other hand, the sequence of sizes of maximal consecutive intervals of Ω (t) read from the element s towards the direction s, s + 1, . . . is the reverse of the sequence of sizes of maximal consecutive intervals of Ω (−t) read from the element s towards the direction s, s − 1, . . . . Hence Σn (t) = Σn (−t). □ Now, in order to find the minimum value of the function Σn (t), by Lemma 3.6 it suffices to restrict t ∈ {1, 3, . . . , n/2}. We will need notations in Definition 3.7. Definition 3.7. Let k ≥ 4 and n ≥ 2k + 2. Suppose that n ≡ 2 (mod 4) and gcd(n, k) = 2. Let k′ be the smallest positive multiplicative inverse of k modulo n/2, namely kk′ ≡ 1 (mod n/2) and k′ < n/2. Define the functions m± (t) = mod (±1 − t)k′ /2, n/2 ,

(

)

[0, ⌊n/4⌋ − m± (t)], if m± (t) < n/4; ± [n/2 − m (t), ⌊n/4⌋], if m± (t) > n/4, ( ) m(t) = min m+ (t), m− (t) , and ( ) M(t) = max m+ (t), m− (t) . ±

{

S (t) =

Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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In terms of the set functions S ± (t), we provide lower bounds of the function Σn (t) in Lemma 3.8, based on which we obtain lower bounds of Σn (t) in terms of n and k in Lemma 3.10. This enables us to express the minimum value of Σn (t) in terms of n and k; see Theorem 3.11. Example 3.14 helps understand Definition 3.7 and all the deductions. In what follows, we omit the variable t in stating the functions Σn (t), m± (t), S ± (t), m(t) and M(t) when there is no risk of confusion. Lemma 3.8. The number Σn has the following lower bounds:

Σn ≥ |S ± | and

(3.4)

Σn ≥ |S | + |S | − |S ∩ S | − |S ∩ (S − k )|, +

−

+

−

+

−

′

(3.5)

where the equality in Eq. (3.5) holds if any one of the following is true:

• S + ∩ S − = ∅; • S + ∩ (S − − k′ ) = ∅; • S + ∩ (S + + k′ ) = S − ∩ (S − + k′ ) = ∅. Proof. Assume that t ∈ {1, 3, . . . , n/2}. Consider the set

{0, 2k, . . . , (n/2 − 1)k, t , t + 2k, . . . , t + (n/2 − 1)k}. It is elementary that these integers are contiguous modulo n. By parity arguments, we can suppose that there exist some 0 ≤ i, j ≤ ⌊n/4⌋ such that the elements 2ki and 2kj + t are consecutive modulo n. Then 2kj + t ≡ 2ki ± 1 (mod n), that is, j ≡ i + m± (mod n/2). From definition, the sets of solutions i± to the above two modular equations are S ± respectively. In other words, every solution i+ ∈ S + corresponds to the consecutive elements 2ki+ and 2ki+ + 1 modulo n, and every solution i− ∈ S − corresponds to the consecutive elements 2ki− and 2ki− − 1 modulo n. Note that every pair of consecutive elements contributes 1 to the sum Σn . The pairs of the former form are distinct, and so are the pairs of the latter form. This implies the desired Eq. (3.4). In order to show Eq. (3.5), we write Te (t) = S + (t) ∩ S − (t),

(3.6)

To (t) = S (t) ∩ (S (t) − k ), +

−

′

and

D (t) = S (t) ∩ (S (t) + k ). ±

±

±

′

(3.7) (3.8)

There are two ways in which consecutive elements may form consecutive sectors of size at least 3. Namely, either (i) 2ki+ ≡ 2ki− (mod n) for some i± ∈ S ± , that is, i+ = i− ∈ Te , or (ii) 2ki+ + 1 ≡ 2ki− − 1 (mod n) for some i± ∈ S ± , that is, i+ ∈ To . We obtain Eq. (3.5) by counting the number of distinct elements. The equality in Eq. (3.5) holds if the size of every consecutive sector is at most 3, namely, the modular equation 2ki ≡ 2kj − 2 (mod n), or equivalently, j ≡ i + k′ (mod n/2), has neither a solution pair (i, j) in S + , nor a solution pair (i, j) in S − . In other words, the equality holds if D+ = D− = ∅. In particular, if Te = ∅, then the size of every consecutive sector is at most 3 and the equality holds. For the same reason, the equality holds if To = ∅. This completes the proof. □ From the definition, we can see that 0 ≤ m < M ≤ n/2 − 1. In fact, m+ − m− =

{

k′ ,

if m+ > m− ,

k − n/2,

if m+ < m− .

′

(3.9)

In Lemma 3.9, we establish technical implications which will be frequently used in what follows. Lemma 3.9. Assume that Σn ≤ M − m. Then m < n/ 4 < M .

(3.10)

Assume that Eq. (3.10) holds. Then

|S + | + |S − | = M − m + 1 and S + ∩ S − = [n/2 − M , ⌊n/4⌋ − m]. Proof. Note that |S ± | = ⌈|m± − n/4|⌉. By Eq. (3.4), we obtain that

|m± − n/4| < Σn ≤ |m+ − m− |.

(3.11)

If m > n/4, then Eq. (3.11) implies M − n/4 < M − m, a contradiction. If M < n/4, then Eq. (3.11) implies n/4 − m < M − m, a contradiction. This proves Eq. (3.10). The remaining results follow immediately from Definition 3.7. □ Lemma 3.10. The number Σn (t) is greater than or equal to min(k′ , n/2 − k′ ). If equality holds, then m(t) < n/4 < M(t) and n/6 < k′ < n/3. Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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Proof. If Eq. (3.10) does not hold, then Lemma 3.9 and Eq. (3.9) imply that

Σn > M − m ≥ min(k′ , n/2 − k′ ). Therefore, we can suppose that Eq. (3.10) holds from now on. From the definition of k′ , we obtain k′ ̸ = n/6 and k′ ̸ = n/3. We continue to use the notation Te , To and D± defined by Eqs. (3.6)–(3.8). Case 1. Suppose that m+ > m− . By Eq. (3.9), we obtain m+ − m− = k′ . When k′ < n/4, we can compute that Te = To = ∅. By Lemmas 3.8 and 3.9, we obtain Σn = |S + | + |S − | = k′ + 1. In this case, the desired inequality is proved and the equality never holds. When k′ > n/4, we obtain To = [n/2 − k′ , ⌊n/4⌋]. By Eq. (3.5) and Lemma 3.9, we find

Σn ≥ |S + | + |S − | − |Te | − |To | = M − m + 1 − (⌊n/4⌋ − m − n/2 + M + 1) − (⌊n/4⌋ − n/2 + k′ + 1) = n/2 − k′ = min(k′ , n/2 − k′ ). Assume that the equality in the above inequality holds, i.e., Σn = n/2 − k′ . By Eq. (3.4) and Lemma 3.9, we deduce that 2Σn ≥ |S + | + |S − | = M − m + 1 = k′ + 1, which implies k′ < n/3. Case 2. Suppose that m+ < m− . In this case, Eq. (3.9) yields m− − m+ = n/2 − k′ . When k′ > n/4, we can compute that Te = To = ∅. By Lemmas 3.8 and 3.9, we obtain Σn = |S + | + |S − | = n/2 − k′ + 1. When k′ < n/4, we can compute that To = [0, ⌊n/4⌋ − k′ ]. By Eq. (3.5) and Lemma 3.9, we obtain

Σn ≥ |S + | + |S − | − |Te | − |To | = M − m + 1 − (⌊n/4⌋ − m − n/2 + M + 1) − (⌊n/4⌋ − k′ + 1) = k′ . On the other hand, by Eq. (3.4) and Lemma 3.9, we deduce that 2Σn ≥ |S + | + |S − | = M − m + 1 = n/2 − k′ + 1. If Σn = k′ , then the above inequality yields k′ > n/6. This completes the proof. □ Theorem 3.11. Let k ≥ 4 and n ≥ 2k + 2. Suppose that n ≡ 2 (mod 4) and gcd(n, k) = 2. For any vertex cover W that minimally covers I ,

Σ (W ) = min(k′ , n/2 − k′ ) + δ,

(3.12)

where k is the smallest positive multiplicative inverse of k modulo n/2 and ′

δ=

0, if n/6 < k′ < n/3;

{

1, otherwise.

Proof. It suffices to show that min

t ∈{1, 3, ..., n/2}

Σn (t) = min(k′ , n/2 − k′ ) + δ,

(3.13)

By Lemma 3.10, the left hand side of Eq. (3.13) is greater than or equal to the right hand side. Thus it suffices to show the existence of a positive odd integer t ≤ n/2 such that the number Σn (t) equals the right hand side of Eq. (3.13). From definition, k′ ̸ ∈ {n/6, n/4, n/3}. Set ⎧ ⌊ n ⌋ ⎪ + 1, if k′ < n/6; 2 ⎪ ⎪ ⎪ 4k′ ⎪ ⌊ ⌋ ′ ⎪ n − 4k ⎪ ′ ⎪ ⎪ ⎨6 12k′ − 2n + 5, if n/6 < k < n/4; ∗ ⌊ ⌋ ′ t = ⎪6 4k − n + 5, if n/4 < k′ < n/3; ⎪ ⎪ ′ ⎪ ⎪ ⌊ 4n − 12k⌋ ⎪ ⎪ n ⎪2 ⎪ + 1, if k′ > n/3. ⎩ 2n − 4k′ It is routine to check that t ∗ is a positive odd integer less than or equal to n/2. Below we verify that Σn (t ∗ ) = min(k′ , n/2 − k′ ) + δ . Case 1. Suppose that k′ < n/6. By the same argument as in Case 1 of the proof of Lemma 3.10, it suffices to check m− (t ∗ ) < n/4 < m+ (t ∗ ). In fact, this inequality holds true since (⌊ n ⌋ ) ⌊ n ⌋ n n + ∗ ′ + 1 and m (t ) = − k . m− (t ∗ ) = − k′ 2 4k′ 2 4k′ Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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Case 2. Suppose that n/6 < k′ < n/4. By the argument in Case 2 of the proof of Lemma 3.10 and by Lemma 3.8, it suffices to check that M(t ∗ ) = m− (t ∗ ) and that D± (t ∗ ) = ∅, where D± (t) are defined by Eq. (3.8). We claim that the inequality n/4 − k′ < m+ (t ∗ ) < 2k′ − n/4

(3.14)

implies the desired relations. In fact, Eq. (3.14) implies m < 2k − n/4 < k . If M = m , then Eq. (3.9) gives m = m − k′ < 0, which is absurd. This proves M = m− . Consequently, Eq. (3.9) reduces to M − m = n/2 − k′ . Together with Eq. (3.14), we can check Eq. (3.10). Let +

′

′

+

−

+

α (t) = [0, ⌊n/4⌋ − m(t)] and β (t) = [n/2 − M(t), ⌊n/4⌋]. By Lemma 3.9 and Eq. (3.14), we obtain D+ = α ∩ mod(α + k′ , n) = [k′ , ⌊n/4⌋ − m] = ∅, D− = β ∩ mod(β + k′ , n) = [n/2 − M + k′ , ⌊n/4⌋] = ∅. This proves the claim. Now we shall verify Eq. (3.14). In fact, m+ (t + 6) − m+ (t) ≡ −s

(mod n/2),

where s = 3k − n/2 is the number of integers contained in the interval (n/4 − k′ , 2k′ − n/4). Since m+ (5) = n/2 − 2k′ > n/4 − k′ , the integer m+ (t0 ) belongs to the desired interval, where ′

⌊ t0 = 5 + 6 ·

m+ (5) − (⌈n/4⌉ − k′ )

⌋

s

= t ∗.

Case 3. Suppose that n/4 < k′ < n/3. By the proof of Lemma 3.10 and by Lemma 3.8, it suffices to check that M(t ∗ ) = m+ (t ∗ ) and that D± (t ∗ ) = ∅. We claim that the inequality 2k′ − n/4 < m+ (t ∗ ) < 3n/4 − k′

(3.15)

implies the desired equations. In fact, Eq. (3.15) implies m > 2k − n/4 > k . If M = m , then Eq. (3.9) gives m− = m+ − k′ + n/2 > n/2, which is absurd. This proves M = m+ . Consequently, Eq. (3.9) reduces to M − m = k′ . Together with Eq. (3.15), we can check Eq. (3.10). By Lemma 3.9 and Eq. (3.15), we obtain +

′

′

−

D+ = β ∩ mod(β + k′ , n) = [n/2 − M , k′ − ⌈n/4⌉] = ∅, D− = α ∩ mod(α + k′ , n) = [0, k′ − m − ⌈n/4⌉] = ∅. Now we shall verify Eq. (3.15). In fact, (mod n/2),

m+ (t + 6) − m+ (t) ≡ s′

where s′ = n − 3k′ is the number of integers in the interval (2k′ − n/4, 3n/4 − k′ ). Since m+ (5) = n − 2k′ < 3n/4 − k′ , the integer m+ (t0 ) belongs to the desired interval, where

⌊ t0 = 5 + 6 ·

(⌊3n/4⌋ − k′ ) − m+ (5) s′

⌋

= t ∗.

Case 4. Suppose that k′ > n/3. By the proof of Lemma 3.10, it suffices to check that m+ (t ∗ ) < n/4 < m− (t ∗ ). In fact, the finite sequence m+ (1), m+ (3), . . . , m+ (n/2), is an arithmetic progression with common difference n/2 − k′ . This sequence contains a term m+ (t0 ) with

⌊n/4⌋ t0 = 2 n/2 − k′

⌊

⌋

+ 1 = t ∗,

which is the maximum t such that m+ (t) < n/4. As an immediate consequence, one has m− (t ∗ ) = m+ (t ∗ ) + (n/2 − k′ ) > n/4. This completes the proof. □ In particular, when k = 4, Theorem 3.11 reduces to Corollary 3.12. Corollary 3.12. Let n ≥ 10 such that n ≡ 2 (mod 4). Then β (n, 4) ≤ ⌈(9n + 4)/8⌉. Proof. Let k = 4 and let W be a vertex cover of P(n, k) that minimally covers the inner cycles. If n ≡ 6 (mod 8), then k′ = (n + 2)/8 < n/6. By Theorem 3.11, we deduce that Σ (W ) = k′ + 1 = (n + 10)/8. Otherwise n ≡ 2 (mod 8); then k′ = (3n + 2)/8 > n/3 and Σ (W ) = n/2 − k′ + 1 = (n + 6)/8. Hence we obtain a proof by β (n, k) ≤ n + Σ (W ) from Lemma 2.7. □ Please cite this article as: D.D.D. Jin and D.G.L. Wang, On the minimum vertex cover of generalized Petersen graphs, Discrete Applied Mathematics (2019), https://doi.org/10.1016/j.dam.2018.12.011.

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Table 3.1 Four examples of t that optimize the function Σn (t). n

k

k′

t∗

m+

m−

S+

S−

Te

To

Σn

90 98 226 150

34 26 84 56

4 17 74 71

11 23 29 19

25 9 94 36

21 41 20 60

[20, 22] [0, 15] [19, 56] [0, 1]

[0, 1] [8, 24] [0, 36] [35, 37]

∅ [8, 15] [19, 36] ∅

∅ [0, 7] [39, 56] ∅

5 17 39 5

Theorem 3.13. Let k ≥ 4 and n ≥ 2k + 2. Suppose that gcd(n, k) = 2 and n ≡ 2 (mod 4). Then β (n, k) = n + 2 if and only if n = 2k + 2. Proof. Since k ≥ 4, we obtain n ≥ 10. Let C be a minimum vertex cover of the graph P(n, k). By Lemma 2.7, we find β (n, k) = n + 2 ⇐⇒ Σ (C ) = 2. Let us retain the notation in Theorem 3.11. Suppose that n = 2k + 2. Since d = 2, we find k is even. On the other hand, from definition, we find k′ = k. It follows that min(k′ , n/2 − k′ ) = 1 and δ = 1. By Theorem 3.11 and Lemma 2.7, we obtain Σ (C ) = 2. Conversely, suppose that β (n, k) = n + 2. By Lemma 3.1 the cover C minimally covers I , which allows us to apply Theorem 3.11. If k′ < n/6, then k′ + 1 = Σ (C ) = 2. Thus k′ = 1 and k = 1, a contradiction. If n/6 < k′ < n/4, then k′ = 2, n = 10, and k = 3, a contradiction. If n/4 < k′ < n/3, then n/2 − k′ = Σ (C ) = 2. Since k′ < n/3, we deduce that n = 10, k′ = 3, k = 2, a contradiction. Hence, we find k′ > n/3. Then n/2 − k′ + 1 = Σ (C ) = 2. Thus k′ = n/2 − 1 and k = n/2 − 1, i.e., n = 2k + 2. This completes the proof. □ Example 3.14. Table 3.1 contains four examples of t that optimize the function Σn (t), for k′ in each of the intervals (0, n/6), (n/6, n/4), (n/4, n/3), and (n/3, n/2), where the symbols Te and To are defined by Eqs. (3.6) and (3.7). 3.3. Conclusion We are now in a position to characterize the generalized Petersen graphs P(n, k) whose minimum vertex cover number is n + 2 under the condition that the 2-adic valuation of n is less than or equal to that of k. Theorem 3.15. Let k ≥ 4 and n ≥ 2k + 2. If the 2-adic valuation of n is less than or equal to that of k, then a minimum vertex cover of the generalized Petersen graph P(n, k) has size n + 2 if and only if n ∈ {2k + 2, 3k − 1, 3k + 1}. Proof. From Lemma 3.1, we see that d ∈ {1, 2}. Combining Theorems 3.4 and 3.13, we obtain Theorem 3.15. □ Here is an immediate corollary. Corollary 3.16. β (14, 4) = 17. Proof. Corollary 3.12 implies β (14, 4) ≤ 17. From Theorem 3.15, we obtain β (14, 4) ≥ 17. Therefore, we obtain β (14, 4) = 17. □ Our method, to use elementary number theory, does not work efficiently when the 2-adic valuation of n is larger than that of k. This is mainly because of the possibility that both the outer cycle and the inner cycles are not minimally covered in a minimum vertex cover. Acknowledgments The authors are indebted to the two anonymous referees and Nick Early for their suggestions leading to increased elegance in the present paper. The second author is grateful to the hospitality of organizers of the second Malta conference in graph theory and combinatorics, and to that of Massachusetts Institute of Technology when he was a visiting scholar there. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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